The Quotient Rule is an important method used for finding the derivative of a function that divides two other functions.

This rule is used when we have a function like ( h(x) = \frac{f(x)}{g(x)} ). We can only use this when ( g(x) ) is not zero, because you can't divide by zero.

How to Use the Quotient Rule

To understand how to use the Quotient Rule, we start with our function ( h(x) ) and use a special way of finding a derivative using limits. We want to find ( h'(x) ):

$h'(x) = \lim_{h \to 0} \frac{h(x + h) - h(x)}{h}$

We now plug our ( h(x) ) into this:

$h'(x) = \lim_{h \to 0} \frac{\frac{f(x + h)}{g(x + h)} - \frac{f(x)}{g(x)}}{h}$

Next, we need to combine the fractions. To do this, we find a common denominator:

$= \lim_{h \to 0} \frac{f(x + h)g(x) - f(x)g(x + h)}{h \cdot g(x + h)g(x)}$

After some math and using L'Hôpital's Rule if needed, we reach a nice formula:

$h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$

This shows us that finding the derivative of a quotient involves looking at both the top (numerator) and bottom (denominator) parts of the fraction closely.

Examples

Let’s see how this works with a couple of simple examples.

Example 1: Finding the Derivative

If we have ( h(x) = \frac{x^2 + 1}{x - 3} ), we can identify:

• ( f(x) = x^2 + 1 )
• ( g(x) = x - 3 )

1. First, we find ( f'(x) = 2x ) and ( g'(x) = 1 ).

2. Next, we plug these into the Quotient Rule formula:

$h'(x) = \frac{(2x)(x - 3) - (x^2 + 1)(1)}{(x - 3)^2}$

3. Then, we simplify the top part:

$h'(x) = \frac{2x^2 - 6x - x^2 - 1}{(x - 3)^2} = \frac{x^2 - 6x - 1}{(x - 3)^2}$

Example 2: Using Trigonometry

Now, let’s consider ( h(x) = \frac{\sin(x)}{\cos(x)} ).

1. Here, we have:

• ( f(x) = \sin(x) )
• ( g(x) = \cos(x) )

2. We find the derivatives:

• ( f'(x) = \cos(x) )
• ( g'(x) = -\sin(x) )

3. Using the Quotient Rule, we get:

$h'(x) = \frac{\cos(x)\cos(x) - \sin(x)(-\sin(x))}{(\cos(x))^2}$

4. After simplifying, we find:

$h'(x) = \frac{\cos^2(x) + \sin^2(x)}{(\cos(x))^2} = \frac{1}{\cos^2(x)}$

This result tells us that ( h'(x) = \sec^2(x) ), which is a familiar function!

Practice Problems

To really understand the Quotient Rule, try these problems on your own:

1. Differentiate ( h(x) = \frac{e^x}{x^2 + 1} ).
2. Find the derivative of ( h(x) = \frac{x^3 - 1}{x + 2} ).
3. Compute ( h(x) = \frac{\tan(x)}{x^2} ).
4. Differentiate ( h(x) = \frac{1}{\ln(x)} ).

These problems will help you practice finding the functions ( f(x) ) and ( g(x) ) and using the Quotient Rule to get the answers.

Using the Quotient Rule gives you a strong way to tackle derivatives in different math problems, making it a key tool in calculus!